Answer :
To determine which expression is a prime polynomial, we need to recognize that a prime polynomial is one that cannot be factored into simpler polynomials using coefficients in the set of integers or rational numbers.
Let's analyze each option:
A. [tex]\( x^3 - 27y^6 \)[/tex]
- This can be recognized as a difference of cubes, since [tex]\( 27y^6 \)[/tex] is [tex]\( (3y^2)^3 \)[/tex]. The formula for factoring a difference of cubes is [tex]\( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)[/tex]. With [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex], this polynomial can be factored as:
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Since it can be factored, it's not a prime polynomial.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the greatest common factor here, which is [tex]\( x \)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
This expression is not prime because it can be factored by taking out [tex]\( x \)[/tex].
C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
This expression can be factored, so it is not a prime polynomial.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This doesn't have any obvious simple factors, but let's attempt to see if it's factorable by simple means:
[tex]\[
x^4 + 20x^2 - 100 = (x^2 + 10)^2 - 200
\][/tex]
- However, it might not readily break down into polynomial with integer coefficients. Despite first views, more advanced polynomial tests can reveal its irreducibility in special cases.
Since none of the given polynomials can obviously showcase they cannot be factored by basic inspection and algebraic formulas, and knowing the context in polynomial types, it is important to rely on deducing certain polynomial property traits from structure which indicates complicated terms might resist simplifying factors.
Given these analyses, none of the expressions is a prime polynomial based on basic factorization principles and typical algebraic factorization.
Thus, the answer obtained is that there is no clear choice that stands significantly as a prime polynomial based on conventional factoring, yet complex forms sometimes are considered in rare educational contexts in deeper polynomial studies.
Let's analyze each option:
A. [tex]\( x^3 - 27y^6 \)[/tex]
- This can be recognized as a difference of cubes, since [tex]\( 27y^6 \)[/tex] is [tex]\( (3y^2)^3 \)[/tex]. The formula for factoring a difference of cubes is [tex]\( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)[/tex]. With [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex], this polynomial can be factored as:
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
Since it can be factored, it's not a prime polynomial.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the greatest common factor here, which is [tex]\( x \)[/tex]:
[tex]\[
x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
This expression is not prime because it can be factored by taking out [tex]\( x \)[/tex].
C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\[
3(x^2 + 6y)
\][/tex]
This expression can be factored, so it is not a prime polynomial.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This doesn't have any obvious simple factors, but let's attempt to see if it's factorable by simple means:
[tex]\[
x^4 + 20x^2 - 100 = (x^2 + 10)^2 - 200
\][/tex]
- However, it might not readily break down into polynomial with integer coefficients. Despite first views, more advanced polynomial tests can reveal its irreducibility in special cases.
Since none of the given polynomials can obviously showcase they cannot be factored by basic inspection and algebraic formulas, and knowing the context in polynomial types, it is important to rely on deducing certain polynomial property traits from structure which indicates complicated terms might resist simplifying factors.
Given these analyses, none of the expressions is a prime polynomial based on basic factorization principles and typical algebraic factorization.
Thus, the answer obtained is that there is no clear choice that stands significantly as a prime polynomial based on conventional factoring, yet complex forms sometimes are considered in rare educational contexts in deeper polynomial studies.
